Tuesday 3 August 2021

Pentatrigesimal

Regarding the previous list which I recently posted, the answer is that the list was prime numbers up to 42,875 listed in base 35. 42,875 is 35 cubed and in base 35 is of course 1000.

The logic behind base 35, is, as you might gather by looking at the table, to write numbers using all numbers and all letters of the English alphabet except O. The reason for not using O is, of course, that it is so similar to zero and that casually zero is often read as “O”. In fact, when I was a child, I was often told that “O is a letter, and zero [0] is the number” whenever I read the number as “O”, but would reply that “O is a number”.

The properties of this system of all numbers and all letters except O, scientifically called “pentatrigesimal” are both familiar and unfamiliar compared to decimal.

Like ten, 35 is a deficient semiprime (=5x7). However, 35 is a much more deficient number than ten, having a deficiency of 22 (or 63 percent) as against 2 for the number 10. Thus, the proportion of fractions that terminate in pentatrigesimal is much smaller than in decimal: only denominators whose only prime factors are 5 and 7 terminate, and there are only 38 such numbers smaller than 1,500,625 (354, or 10000 in pentatrigesimal):

Decimal

Pentatrigesimal

Prime factorisation

5

5

5

7

7

7

25

Q

5*5

35

10

5*7

49

1E

7*7

125

3K

5*5*5

175

50

5*5*7

245

70

5*7*7

343

9T

7*7*7

625

HV

5*5*5*5

875

Q0

5*5*5*7

1,225

100

5*5*7*7

1,715

1E0

5*7*7*7

2,401

1YL

7*7*7*7

3,125

2JA

5*5*5*5*5

4,375

3K0

5*5*5*5*7

6,125

500

5*5*5*7*7

8,575

700

5*5*7*7*7

12,005

9T0

5*7*7*7*7

15,625

CRF

5*5*5*5*5*5

16,807

DQ7

7*7*7*7*7

21,875

HV0

5*5*5*5*5*7

30,625

Q00

5*5*5*5*7*7

42,875

1000

5*5*5*7*7*7

60,025

1E00

5*5*7*7*7*7

78,125

1TS5

5*5*5*5*5*5*5

84,035

1YL0

5*7*7*7*7*7

109,375

2JA0

5*5*5*5*5*5*7

117,649

2R1E

7*7*7*7*7*7

153,125

3K00

5*5*5*5*5*7*7

214,375

5000

5*5*5*5*7*7*7

300,125

7000

5*5*5*7*7*7*7

390,625

93VQ

5*5*5*5*5*5*5*5

420,175

9T00

5*5*7*7*7*7*7

546,875

CRF0

5*5*5*5*5*5*5*7

588,245

DQ70

5*7*7*7*7*7*7

765,625

HV00

5*5*5*5*5*5*7*7

823,543

J79T

7*7*7*7*7*7*7

1,071,875

Q000

5*5*5*5*5*7*7*7

1,500,625

10000

5*5*5*5*7*7*7*7

As for primes with short recurring periods, there are 37 primes with pentatrigesimal periods of 20 or shorter, vis-à-vis 32 in decimal. One of these is as long as twenty decimal digits, another eighteen, and another sixteen, so I will not replicate all numbers with pentatrigesimal periods of 20 or shorter here. There are twenty primes with pentatrigesimal periods of 12 or shorter, vis-à-vis sixteen in decimal:

Pentatrigesimal

Pentatrigesimal repetend

Period

Decimal

2

0.Ḣ

1

2

H

0.2̇

17

3

0.ḂṄ

2

3

D

0.2̇P8̇

3

13

2S

0.0̇CṀ

97

HI

0.0̇1ZẎ

4

613

W

0.1̇4I29̇

5

31

15NR

0.0̇00V4̇

49,831

BC

0.0̇32ZWẊ

6

397

18

0.0̇TH38YḊ

7

43

UBENM

0.0̇00016Ṡ

44,007,727

HHHI

0.0̇001ZZZẎ

8

750,313

J

0.1̇UGK97CWḂ

9

19

1UGM3P

0.0̇0000IZZĠ

96,753,079

B

0.3̇6CQFWTM9J̇

10

11

339G

0.0̇00BAZZZNṖ

132,631

N

0.1̇I94JSDPC63̇

11

23

1AM8

0.0̇00RUI4JFCḊ

55,903

171RAE4

0.0̇00000U4LMṘ

2,208,546,869

7X

0.0̇4ESEMZVK7KĊ

12

277

4EN

0.0̇07X7WZZS2S3̇

5,413

Pentatrigesimal fractions for the first sixty natural numbers (periods listed in decimal notation) are:

 

Pentatrigesimal reciprocal

Period

Decimal

1

1

0

1

2

0.Ḣ

1

2

3

0.ḂṄ

2

3

4

0.8̇Ṙ

2

4

5

0.7

0

5

6

0.5̇U̇

2

6

7

0.5

0

7

8

0.4̇Ḋ

2

8

9

0.3̇Ẇ

2

9

A

0.3Ḣ

1

10

B

0.3̇6CQFWTM9J̇

10

11

C

0.2̇Ẋ

2

12

D

0.2̇P8̇

3

13

E

0.2Ḣ

1

14

F

0.2ḂṄ

2

15

G

0.2̇6JṖ

4

16

H

0.2̇

1

17

I

0.1̇Ẏ

2

18

J

0.1̇UGK97CWḂ

9

19

K

0.1Ṙ8̇

2

20

L

0.1ṄḂ

2

21

M

0.1̇KNV7YEB4Ṡ

10

22

N

0.1̇I94JSDPC63̇

11

23

P

0.1̇Ġ

2

24

Q

0.1E

0

25

R

0.1̇C4̇

3

26

S

0.1̇ACYPṀ

6

27

T

0.18̇Ṙ

2

28

U

0.1̇78FP4TYSRJAV6̇

14

29

V

0.15̇U̇

2

30

W

0.1̇4I29̇

5

31

X

0.1̇39UIKSĊ

8

32

Y

0.1̇248GYXVRİ

10

33

Z

0.1̇

1

34

10

0.1

0

35

11

0.0̇Ż

2

36

12

0.0̇Y3SF4QIX5NMPKTD8HZ1W7JV9G2UBCAE6LRḢ

36

37

13

0.0̇X8A4L6FṄ

9

38

14

0.0̇WECJṘ

6

39

15

0.0V̇L̇

2

40

16

0.0̇UVQLBYA8ISB3EHXFCT5Z549DN1PRG7NWKH2JM6U̇

40

41

17

0.0U̇5̇

2

42

18

0.0̇TH38YḊ

7

43

19

0.0̇SUF3Z75JẆ

10

44

1A

0.0Ṡ7̇

2

45

1B

0.0̇RM29W6UNKJ̇

11

46

1C

0.0̇R286PK3QB5YHVILKUSJCN2Z8XRTAEW9NU1H4GDE57FMBẊ

46

47

1D

0.0̇QI8̇

4

48

1E

0.0Q

0

49

1F

0.0PḢ

1

50

1G

0.0̇Ṗ

2

51

1H

0.0̇NJI62̇

6

52

1I

0.0̇N3YNS2MFUQ3AJTDVD798KGHU1ZBW1B7XCJ59WPF6L4LSQREIH5Ẏ

52

53

1J

0.0̇MNZCḂ

6

54

1K

0.0Ṁ9J36CQFWṪ

10

55

1L

0.0L̇V̇

2

56

1M

0.0̇LH6RE4AFC9TV32FYṠ

18

57

1N

0.0̇L47UJWZDVS5Ḟ

14

58

1P

0.0̇KRPB9H745BV8WFEU2D1SA2YTGLCĠ

29

59

1Q

0.0K̇Ė

2

60

As you can see, the periods are different from those in decimal, apart from powers of five, the full period prime 47 (and 61 if I extended the list), the number 52, and the odd semiprimes 39 and 57. Noteworthy is the large number of denominators with pentatrigesimal period 2: nineteen numbers out of sixty, vis-à-vis only five in decimal.

2 comments:

123 said...

I think that not skip the letter O, base 36 is better, by your logic O and 0 are confused, why not I and 1? Or S and 5?

See these pages:

http://www.tonymarston.net/php-mysql/converter.html, https://www.dcode.fr/base-36-cipher, http://www.urticator.net/essay/5/567.html, http://www.urticator.net/essay/6/624.html, https://docs.python.org/3/library/functions.html#int, https://numpy.org/doc/stable/reference/generated/numpy.base_repr.html, https://reference.wolfram.com/language/ref/BaseForm.html, https://support.microsoft.com/en-us/office/base-function-2ef61411-aee9-4f29-a811-1c42456c6342, https://www.cut-the-knot.org/recurrence/word_primes.shtml, https://oeis.org/A072922, https://oeis.org/A073421, https://oeis.org/A002488 (the Alonso del Arte comment in Jul 01 2012), https://en.wikipedia.org/wiki/Base36, https://web.archive.org/web/20150320103231/https://en.wikipedia.org/wiki/Base_36, https://fr.wikipedia.org/wiki/Syst%C3%A8me_%C3%A0_base_36 (in French), https://ja.wikipedia.org/wiki/%E4%B8%89%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95 (in Japanese), https://baseconvert.com/, https://baseconvert.com/high-precision, https://www.calculand.com/unit-converter/zahlen.php?og=Base+2-36&ug=1, http://www.unitconversion.org/unit_converter/numbers.html, http://www.unitconversion.org/unit_converter/numbers-ex.html, http://www.kwuntung.net/hkunit/base/base.php (in Chinese), https://linesegment.web.fc2.com/application/math/numbers/RadixConversion.html (in Japanese)

jpbenney said...

I can see your point, but I think it is much easier to confuse O and 0 than they other two. At all events, as a child I would read "0" as "oh", but never the same with "1" and "I" (except when reading Roman numerals in my old World Cars 1984 and World Cars 1985 books.